Convergence of a branching type recursion

A NNALES DE L ’I. H. P., SECTION B

M. C ^RAMER

L. R ^ÜSCHENDORF

Convergence of a branching type recursion

Annales de l’I. H. P., section B, tome 32, n

^o

6 (1996), p. 725-741

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Convergence ^of

^a

branching type ^recursion

M. CRAMER and L.

RÜSCHENDORF

University ^of Freiburg, ^{Institut fur} Mathematische Stochastik,

Hebelstr. 27, Freiburg 79104, Germany.

Vol. 32, n° 6, 1996, p. 725-741 Probabilités et Statistiques

ABSTRACT. - The

asymptotic

distribution of a

branching type

^recursion is

investigated.

The recursion is

given by XiL(i)n-1

⁺

^Y,

where

( X L )

^{is a} random sequence,

( L ~L ~ 1 )

^{are iid}

^copies

^of

^Ln-l,

^{K is a}

random number and

.~, ( L ~ z ~ 1 ) , ~ ( X i ) , Y ~

^are

independent.

This recursion has been studied

intensively

in the literature in the case that

0,

K is nonrandom and Y ⁼ 0. Included in the more

general

^{recursion are}

branching

processes, ^a model of Mandelbrot

( 1974)

^for

studying turbulence,

which was also

investigated

in connection with infinite

particle systems

^and the

expansion

^{of total mass in} the construction of random multifractal

measures. The

stability

in the recursion arises from the fact that it includes

a

smoothing part (addition)

^{and on} the other hand a

part increasing

^the

fluctuation

(random multiplier,

random number and random

immigration).

Our treatment of this recursion is based on a contraction

technique

^which

applies

^{under some} restrictions on the first two moments of the involved random variables. We obtain a

quantitative approximation

^{result. The}

exponential

^{rate of} convergence ^can be observed

empirically. Typically

^after

8 iterations the

limiting

distribution of the recursion is well

approximated.

Key ^words: Branching type recursion, contraction method.

RESUME. - On etudie la distribution

asymptotique

d’une relation de recurrence de

type

branchement donnée par ⁺ Y.

Ici

(Xi)

^{est une} suite de variables aléatoires

réelles,

^sont des

v.a.

indépendantes

de la meme distribution que

L.,,-i,

^{K un nombre}

AMS Classification : ^{60 F} 05, ^{68 C 25.}

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques - ^0246-0203

726 M. CRAMER AND L. RUSCHENDORF

aléatoire et

K, (L~2~ 1 ) , ~ (XL ), Y~

^sont

indépendantes.

Dans la littérature

on a etudie cette recurrence dans le cas où

Xi

^>

0,

K non-aléatoire et Y ⁼ 0. Entre autres la relation de recurrence

plus générale

décrit des processus de

branchement,

^un model de Mandelbrot

(1974)

pour l’étude de

turbulence,

^et

1’ expansion

^{de masse totale} dans la construction des

mesures aléatoires multifractales. La stabilité dans la recurrence vient du fait

qu’elle

^{contient une}

partie

^de

lissage (addition)

^et conversement une

partie augmentant

la fluctuation

(multiplicateur aléatoire,

^nombre

aléatoire et

immigration aléatoire). L’ analyse

^{dans cet article est fondée}

sur une

technique

de contraction

applicable

^sous

quelques

restrictions ^aux deux

premiers

^{moments des v.a.} données. Nous obtenons des résultats

d’ approximation quantitatifs.

^La

rapidité

de convergence

exponentielle

^est

observable

empiriquement. Après

environ 8 iterations la distribution limite de la relation de recurrence est bien

approchée.

1. INTRODUCTION

Consider the

following

recursive sequence

(Ln)

^defined

by

where are iid

copies

^of

Ln,_ 1,

^{is a real} random sequence, K a random number in

No

^{and Y a random}

immigration

^{such that}

K, ~ ( Xi ) . Y ~ . ( L~? ~ 1 )

^are

independent. =

^denotes

^equality

in distribution.

(1)

^{induces a} transformation T on

M1,

^{the set of}

probability

distributions

on

(R1, 31 ) , by letting T( )

be the distribution of

03A3Ki=1 ^{X ; 2i,}

⁺

^V,

^where

(Zi)

^{are iid}

-distributed, (Zi), {(Xi), Y}, K independent.

Some

special

^cases of this transformation resp. recursion have been studied

intensively

in the literature. If 1 then

(1)

^{describes a Galton-}

Watson process with

immigration

Y and the number of descendants of ^an individuum described

by

^K.

(1)

^can be considered from this

point

^{of view}

as a

branching

process with random

multiplicative weights.

^The

special

^case

where K is constant, Y ⁼

0. (Xi )

^{iid and non}

negative

has been introduced

by

Mandelbrot

(1974)

^to

analyse

^a model of turbulence of

Yaglom

^and

Kolmogorov.

^{This case} has been studied

by

Kahane and

Peyrière (1976)

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

and Guivarch

( 1990)

^who considered the

question

^of nontrivial fixed

points of T,

^{existence of moments of the fixed}

points

and convergence of

(L~z ) .

^For

Xi -

the solutions of the fixed

point equation Z a

are Paretian stable distributions

(if 0).

^{For this reason} the solutions

are called semi-stable in Guivarch

( 1990).

^In this paper ^we will be

mainly

interested in the case of

multipliers Xi

and solutions

Zi

^{with moments}

of some order > 2. While the

analysis

^{of Kahane and}

Peyrière (1976)

^is

based on an associated

martingale,

^Guivarch

( 1990)

uses a more

elementary martingale property together

^{with a}

conjugation

relation and moment

type

estimates for the

Lp-distance,

^{0 p 1.}

Motivated

by

^some

problems

^{in infinite}

particle systems Holley

^and

Liggett (1981)

and Durrett and

Liggett (1983)

considered this kind of

smoothing

transformation with

(Xi)

^not

necessarily independent

^and

0,

K constant, ^{Y =} 0. In the last mentioned _paper a

complete analysis

^{of this case could be}

given.

^In

particular

^a necessary and sufficient condition for the existence and the characterization of

(all)

^fixed

points

and a

general

sufficient condition for convergence ^was derived as well as a

generalization

of the result of Kahane and

Peyrière

^{on the existence of}

moments. The method of the paper of Durrett and

Liggett

^{is based on an}

associated

branching

random walk.

The use of contraction

properties

of minimal

Lp-metrics

in this paper allows to obtain

quantitative approximation

results for the recursion

(1).

Under the moment

assumptions

used in this paper the recursion converges

exponentially

^{fast to the}

limiting

distribution. This is demonstrated

by

simulations for several

examples.

Also it is

possible

^{to dismiss with}

the

assumption

^of

nonnegativity,

^to deal with a random number and with

immigration

^Y. This allows to include

branching

process

applications

as well as the consideration of the

development

^{of total mass in the}

construction of multifractal measures

(of

e.g. Arbeiter

(1991)).

^This

example

was

motivating

^{the work on} this paper. After

finishing essentially

^the

investigation

^on this paper ^{we were} informed

by

G. Letac about the

history

of the

problem.

^{We are}

grateful

^to him for his remarks and indications.

In section 2 we consider the case Y -

0,

^{discuss a} robustness

property

^of the

problem,

^{relations to}

previous

^{work and}

give

^{some numerical}

examples.

In section 3 we consider an extension to the case with

immigration

^under

assumptions

^{on the}

X,,

^{Y and}

Lo assuring

^the

stationarity

of the first two moments. The method in this paper is based ^{on a} contraction method w.r.t.

suitable metrics as

developed

^{in a} sequence of further

examples

^{in Rachev}

and Ruschendorf

(1991).

728 M. CRAMER AND L. RUSCHENDORF

2. BRANCHING TYPE RECURSION WITH MULTIPLICATIVE WEIGHTS

In this section we consider the recursion

(1)

^with

possibly dependent multipliers Xi

L and

immigration

^{Y -}

0,

^{i. e.}

where

L~2~ 1

^{are iid}

^copies

^of

^{(X L )}

^{is a square}

integrable

^real

random sequence, K ^a random number in

No

^and

I~, ( X Z ) , (L~~)

^are

independent.

To determine the correct normalization of

( L,~ )

^{we at} first consider the first moments of

(Ln). ^{Denote in}

^:=

ELn,

^{c := :=}

PROPOSITION 2.1. -

lo

⁼

1, In

^{= C".}

Proof - Using

^the

independence assumption

ⁱⁿ

(2)

and conditional

expectations

^{we obtain}

Similarly,

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

In the case b ⁼

0,

vn ⁼ 0 for all n. Therefore we consider

only

^the

case b > 0.

From

(3)

^we obtain that for a

c2,

is of the same order as This makes it

possible

^{to use the}

simple

normalization

by In.

Define for

c ~

⁰

then

ELn =

^{1 and}

Var(Ln)

^--~

2

satisfies the modified recursion

where :=

L(i)n-1 cn-1.

Define

~2

^{to be the set} of distributions on

(tRB~)

with finite second moments and first moment

equal

^{to one} and define T :

Ds

^2014~

~2 by

730 M. CRAMER AND L. RÜSCHENDORF

where

(Zi)

^are i.i.d. random variables with distribution

G, (X;, ) , ( 2; ) . K independent.

^Let

l2

denote the minimal

L2-metric

^on

D2

^defined

by

where

F, G

^are the distribution functions of

respectively.

^{If a}

c2,

then T is a contraction w.r.t.

l ~ .

PROPOSITION 2.2. - Assume that a

c’,

^then

for F,

^{G E}

D2

Proof -

^Let

U ~ l > ‘~ _{F, ~,}

~, E

~,

be choosen ^on

( S~ , ^A, P )

^{such that}

Il~~l~ _ V~1>II~ _ ~~(F’, ~).d2

^and

K, (~~Tt), (Utl~, ~~1~), (U~2~. v~~2~), ...

independent.

^Then

As consequence of

Proposition

^{2.2 T has}

exactly

^{one fixed}

point

ⁱⁿ

D2

with variance

equal

^to

b/ ( c~’

^-

a).

^{The fixed}

point equation

^is

given

ⁱⁿ

Annales de l’Institut Henri Poincaré - Probabilites et Statistiques

terms of random variables

Z, Z,

^E

D2 , Z, (Zi) independent, by

and we obtain as

corollary:

In

particular Ln

converges ⁱⁿ distribution to Z.

PROPOSITION 2.4. -

If

^{K is constant and}

c~

V 2

k h then x.

~

^~~

~

Proof - Ln

^{can be}

equivalently represented by Y,L

^{of the}

following

^form

where

(X~l,....~h.-l,l ; ~~~, X~1,...,~~._l,h ) =(Xl, ~~~, Xh ) ^(c/: Guivarch, 1990).

(Yn )

^{is a}

martingale

^{and therefore is a}

submartingale. Representing

^the

Y,z

in the recursive

way Yn = ~ ~ ~ 1

^where

^{Y~~~ ~l}

^are

independent copies

^of

Y,,-i,

^{one obtains}

One can deduce from Theorem 2.3 that is

uniformly

^{bounded for}

k 2.

By

^induction

over 1~ ~

^{one sees} that the lower order terms in the above

equation

^are

uniformly bounded,

say

by

^{C. Since}

one obtains

732 M. CRAMER AND L. RUSCHENDORF

Therefore,

^the

assumptions

^{of this}

proposition

^{ensure that is}

uniformly

bounded for all k h. The

submartingale

convergence theorem

now

yields

the existence of an

integrable

^{almost sure limit}

of

^Since

Yn

the weak limit Z of

L.n

^is

absolutely h-integrable.

^D

For solutions of the

stationary equation (9)

^{it is}

possible

^{to obtain a}

stability

^{result in terms}

of lp

metrics defined as in

(7)

^{with 2}

replaced by

p.

Suppose

^{we want to}

approximate

the solution S of the

equation

by

the solution of the

"approximate" equation

where we assume

w.l.g.

^{c = 1} and consider the case of

independent

sequences

( X i ) , ( X i ) ,

^{such that}

( X i ) , ( si )

^and

( X.L ) , ( Si )

^are

independent

and K is constant.

PROPOSITION 2.5. -

If

^constant,

~,K 1 l p ( X i , X i )

^{~ and}

~~ 1 ~~Xi~~p ^l,

^then

Proof -

^{From the} definition of

S,

^S*

Annales de l’lnstitut Henri Poincaré - Probabilités et Statistiques

This

implies

^that

A similar idea for

establishing

robustness of

equations

^can be found in Rachev

(1991), Chapter

^19.3.

For random K we

replace Proposition

^2.5

by

Proof - By

^the

triangle inequality

^{and the}

independence assumption

^and

the

assumption ^EXi

⁼

EX,L

Therefore,

Remark. -

(a)

^{In the case of constant K} and

nonnegative Xi

Durrett and

Liggett (1981) proved

^{that the}

stationary

^{solution Z of}

(9)

^{has moments}

of order

j3

^{if and}

only

^if

For

j3

⁼

2, ( 14)

^is

equivalent

^to the condition a

c2

used in Theorem 1. In this sense this condition is

sharp

^when

using l2-distances.

^Guivarch

(1990)

has shown how to dismiss with the second moment

assumption.

734 M. CRAMER AND L. RUSCHENDORF

(b)

For the normalized recursion

(5)

^with

(Xi ) ^i.i.d.,

^{K constant}

(where

we assume

w.l.g.

^{c =}

1),

we can use the

explicit

^form

(cf.

^the

proof

^of

Proposition 2.4)

where are

independent

^and distributed as

Xi,

i.e.

Ln

^{is the}

sum over

product weights

^{in the}

complete K-ary

^{tree. For}

nonnegative multipliers Xi

^{one can} consider further functionals as e.g.

the ^max

product

^{over all}

paths

^of

length

^n.

Taking logarithms

and

application

^of

Kingman’s

subadditive

ergodic

^theorem

yields

^{for some}

constant

/3

This shows that in some sense the max

product weight

^{is not}

larger

^{in the}

order than the average

product weight.

^For some cases

,~~

^is

known,

e.g. for

X; a U ~0,1~ , ,~ ~

^-0.23196

(cf.

^Mahmoud

(1992),

p.

165).

(c)

^For some cases

explicit

solutions of

(9)

^{are known.}

1 )

^If K is constant,

e X L d ~~ ( I , a - h )

^is

Beta-distributed,

^then

Z ~ r(a, /~)

^is Gamma-distributed

(cf. Guivarch, 1990).

2)

^If

Xi 2L,

^{then with}

(Yi ) ^i.i.d., X d _X1Z1

holds

Conversely,

^if

~ h 1 ^{Y; X 1,}

^{then with}

^{(X.? ) i.i.d.,}

the

( Zl )

^solve

(cf Durrett

^and

Liggett (1981)).

3)

^If

( ZL )

^solve

03A3Ki=1 ^{X ; Zi , X,}

^{i >}

^{0, then Yi}

⁼

Z1/03B8iWi, 0

^03B8

2, WL

stable rv’s of index

~,

^solve

Annales de 1’Institut Henri Poincaré - Probabilités et Statistiques

To prove

(18), Z1/03B81W1 =

Yi.

^This

interesting

transformation

property

is used in Guivarch

(1990)

to reduce the case of moments of

Xi

of low order to the case with moments of

higher

^order.

4)

^If

~~ 1 X2 - c2 ~ ^0,

^then

^{Z ~ ~2) normally}

distributed solve

(9).

5)

If Z solves

(9)

^{and Z is an}

independent

copy of

Z,

^{then Z* := Z - Z}

solves

whereXi

^{= and}

^{the Ti}

^are

arbitrary

^random

signs.

^In

particular

^the

case K ⁼

2, Xi a U~-1, l~ independent

is solved

by

^{Z* :- Z - Z where}

Z a rC2~ 2 ~ o).

^-

(d)

^The

following

simulations

(Figures

^{1 and}

2)

^of

Ln, K

⁼

2, Xi,X2 independent, X 1 ~ X 2 d U ~0,1 ~ , respectively X i ~ ~3 ( 2, 2 )

^show

good

coincidence with the theoretical Gamma-distribution.

(e)

^{In the case K =}

2, Xl, X2 independent, X 1 a X? ~ U ~- g , g ~

^no

explicit

solution of

(9)

is known. Nevertheless the

following

simulations

(Figure 3)

^{show that}

Ln

converges very fast ^to the fixed

point

^of

(9).

^The

empirical

distribution functions of

Lio

^and

L12

^can

hardly

^be

distinguished.

Therefore

they

^{may be}

regarded _as

the limit distribution function. The

empirical

distribution function of

L6

^is

already

very close ^to it.

736 M. CRAMER AND L. RUSCHENDORF

(/) Branching

processes. -

Equation (2)

^includes the Galton-Watson process ^as

special

^{case. A} Galton Watson process is defined

by

the recursion

where X ^are i.i.d. with

reproduction

distribution in

No.

^Define

K ~

X and

Xi - 1,

then

for all n.

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

The

equality

^{can be seen}

by

^{induction in n. First}

Zo

⁼

Lo

^{= 1. If}

Zk == Lk

^{for k} n, then

The

assumption a ^cz

^is

equivalent

^{to the} condition EX > 1. D From

equality (20)

^{one can derive}

explicit stationary

distributions and extinction

probabilities

ⁱⁿ some cases. If e.g. X is

geometrically distributed,

P(X = k)

⁼

p(l -

^E

No,

^then

and

Var(X)

^{= The} normalized Galton-Watson _process

Z’ Zn Var

^Zn)

converges ^{to a}

(unique)

solution of the fixed

point equation

The extinction

probability

^is

easily

^{seen to be}

1 PP .

For the normalized continuous

part

^an

equation

^{identical to}

(21 ) (but

with different

variances)

holds. It is well known that this

equation

is solved

by

^the

geometric

^stable

distribution of order

1,

^{i.e. the}

exponential

distribution. This

implies finally

since

3. A RANDOM IMMIGRATION TERM In this section we admit an additional

immigration

^term.

738 M. CRAMER AND L. RUSCHENDORF

Y}, ^K, L { 1 ~ 1, L ~ 2 ~ 2 ; ...

^are

independent, X, and

^{Y have finite}

second moments. The

analysis

^of

(23)

^is

essentially simplified

^{if we assume}

for

lo

^:=

ELo,

vo ^.-

Var(Lo),

If c ⁼

1,

^{then EY =} 0 and

lo arbitrary.

LEMMA 3.1. - Under

assumption (24)

^holds

Proof -

^From

Condition

(24)

^can be achieved with a two

point

distribution for

Lo.

^It

allows to use the

technique

^of

proof

of section 2. A

change

of the initial condition leads to the

necessity

^to

change

the method of

proof

^{and leads}

to a

great variety

of different ^cases to be considered.

We, therefore,

^restrict

to

(24)

in this paper.

As in section 2 ^we introduce the

operator

Annales de l’Institut Henri Poincaré - Probabilités et Statistiques

where

M ( lo, vo)

^{is the set of} distributions with mean

lo

^{and variance} r~

and

(Vi)

^are

^i.i.d., G, (VZ), {(X~).Y~.

^K

independent.

Similarly

^{as in}

Proposition

^{2.2 we can show} the contraction

which

implies

the convergence of

Ln

^{to the}

unique

^fixed

point

^{of T in}

M ( l o, vo )

^{w.r.t. the}

l2 -metric,

the contraction factor

being ~.

We obtain a

sharper

^result

(i. e.

^a smaller contraction

factor) by

^{the use}

of the

Zolotarev-metric (r

instead of

l2.

It is defined

by

PROPOSITION 3.2.

Proof -

^Note

that (r

is ideal of order r w.r.t.

summation,

^{i. e.}

for Z

independent

^of

^X,

^{Y and}

For

(Z;);

^i.i.d. distributed

according

^to

F,G

^we have with X ⁼

(Xz)

740 M. CRAMER AND L. RÜSCHENDORF

Note that

for

our recursion defined

by

^T the first two moments are

conserved.

Therefore,

^{we can}

apply (29)

^with

r

3 and obtain as

corollary:

implies

where Z is a

fixed point of

^{T in}

M(lo; vo).

In

particular ^Ln

converges in distribution to Z.

So also in the case with

immigration

^{one obtains an}

exponential

^rate

of convergence. As ^a consequence after ^a few iterations the

limiting

distribution is well

approximated.

Consider the

following example: L0 = 1 10 ^8-5 -f- 5 ⁸⁰ -+- 2 b2, K

⁼

^2,

Xi. X2 X d X 2 ~ U [- 2 , ] ^{Y ~} ~-i

⁺

~o + ~ ~2.

In this situation

(24)

is fulfilled. The fast convergence is confirmed

by

the closeness of the

empirical

distribution functions of

L6

^and

L8

^{in the}

following

simulation.

FIG. 4. - Empirical distribution functions for Le ^and Ls .

Annales de l’Institut Henri Poincaré - Probabilités ^et Statistiques

REFERENCES

[1] M. ^A. ARBEITER, Random recursive constructions of self-similar fractal measures. The non-compact ^case. Prob. Th. Rel. Fields, Vol. 88, 1991, pp. 497-520.

[2] R. DURRETT and M. LIGGETT, ^Fixed points ^{of the} smoothing transformation.

Z. Wahrscheinlichkeitstheorie verw. Gebiete, Vol. 64, 1983, pp. 275-301.

[3] Y. GUIVARCH, Sur une extension de la notion de loi semi-stable. Ann. Inst. H. Poincaré, Vol. 26, 1990, pp. 261-286.

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(Manuscript ^{received November} 8, 1994;

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